Finite Group Databasebeta

← #509×D127 #30651850

Subgroups — #509×D127 #30651850

Order 16256.

400 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#135×D127C211yesmaximal, characteristic, complemented
2C2×D8×D127C211yesmaximal, characteristic, complemented
3#22550314C211yesmaximal, characteristic, complemented
4#22550313C212yesmaximal, complemented
5#22550312C211yesmaximal, characteristic, complemented
6#22550311C211yesmaximal, characteristic, complemented
7#22550310C212yesmaximal, complemented
8SD32×D127C212yesmaximal, complemented
9D16×D127C212yesmaximal, complemented
10M32×D127C211yesmaximal, characteristic, complemented
11#509×C127C211yesmaximal, characteristic, complemented
12#41×D127—21no—
13C2×D4×D127—21no—
14#16861232C2211yescharacteristic, complemented
15#16861231C2211yescharacteristic, complemented
16#16861230—21no—
17#16861229—21no—
18#16861228C2211yescharacteristic, complemented
19Q16×D127C2211yescharacteristic, complemented
20SD16×D127—21no—
21C2×D1016C2211yescharacteristic, complemented
22C2×#14523233—21no—
23D8×D127—21no—
24D8×D127—21no—
25D8×D127C2212yes—
26D8×D127C2212yescomplemented
27D8×D127C2211yescharacteristic, complemented
28C2×C8×D127C2211yescharacteristic
29#16861154C2212yescomplemented
30#16861153C2212yescomplemented
31#16861152C2212yescomplemented
32D2032C2212yescomplemented
33#16861150C2212yescomplemented
34#16861150C2212yescomplemented
35#16861149C2211yescharacteristic, complemented
36#16861148C2212yescomplemented
37C16×D127C2212yescomplemented
38#135×C127C2211yescharacteristic, complemented
39C2×D8×C127C2211yescharacteristic, complemented
40SD32×C127C2212yescomplemented
41D16×C127C2212yescomplemented
42M32×C127C2211yescharacteristic, complemented
43C23×D127—41no—
44#14523249—21no—
45#14523248—21no—
46#14523248—41no—
47#14523247—21no—
48#14523247—41no—
49Q8×D127—21no—
50C2×D508—21no—
51C2×C22⋊D127—41no—
52D4×D127—22no—
53D4×D127—22no—
54D4×D127—21no—
55D4×D127—41no—
56D4×D127—41no—
57D4×D127—41no—
58C2×C4×D127—41no—
59C2×C4×D127D411yescharacteristic
60Dic508C2311yescharacteristic, complemented
61#14523238—21no—
62#14523237—21no—
63#14523236—21no—
64#14523235—21no—
65D1016—21no—
66D1016C2312yes—
67D1016C2311yescharacteristic, complemented
68#14523233—22no—
69#14523233—22no—
70#14523233—21no—
71C2×C127⋊C8D411yescharacteristic
72C8×D127D411yescharacteristic
73C8×D127D411yescharacteristic
74C8×D127C2311yescharacteristic
75C8×D127C2311yescharacteristic
76C127⋊C16C2312yescomplemented
77#41×C127—21no—
78C2×D4×C127—21no—
79Q16×C127C2311yescharacteristic, complemented
80SD16×C127—21no—
81D8×C127—21no—
82D8×C127C2312yes—
83D8×C127C2311yescharacteristic, complemented
84C2×C8×C127C2311yescharacteristic
85C16×C127C2312yescomplemented
86C22×D127—41no—
87C22×D127—41no—
88C22×D127—42no—
89C22×D127—42no—
90C22×D127—84no—
91C22×D127D811yescharacteristic
92Dic254—21no—
93Dic254—41no—
94D508—22no—
95D508—21no—
96D508—41no—
97D508—41no—
98C22⋊D127—41no—
99C22⋊D127—41no—
100C22⋊D127—42no—
101C22⋊D127—42no—
102C22⋊D127—41no—
103C2×Dic127—41no—
104C2×Dic127D811yescharacteristic
105C4×D127—41no—
106C4×D127—41no—
107C4×D127—41no—
108C4×D127D811yescharacteristic
109C4×D127D811yescharacteristic
110C4×D127C2×D411yescharacteristic
111C4×D127C2×D411yescharacteristic
112C127⋊C8C2×D411yescharacteristic
113C127⋊C8C2×D411yescharacteristic
114C23×C127—41no—
115Q8×C127—21no—
116D4×C127—22no—
117D4×C127—21no—
118D4×C127—41no—
119D4×C127—41no—
120C2×C4×C127—41no—
121C2×C4×C127C2×D411yescharacteristic
122C8×C127C2×D411yescharacteristic
123C8×C127C2411yescharacteristic
124C22×C127—41no—
125C22×C127—42no—
126C22×C127—82no—
127C22×C127C2×D811yescharacteristic
128C2×D127—22no—
129C2×D127—41no—
130C2×D127—42no—
131C2×D127—84no—
132C2×D127—82no—
133C2×D127—82no—
134C2×D127—84no—
135C2×D127C2×D811yescharacteristic
136C2×D127C2×D811yescharacteristic
137C4×C127—41no—
138C4×C127C2×D811yescharacteristic
139C4×C127C22×D411yescharacteristic
140Dic127—41no—
141Dic127C2×D811yescharacteristic
142Dic127C2×D811yescharacteristic
143C2×C127—21no—
144C2×C127—82no—
145C2×C127—81no—
146C2×C127C22×D811yescharacteristic
147D127—21no—
148D127—81no—
149D127—82no—
150D127#50912yesdirect product factor
151C2×#509—1271nomaximal, Sylow
152C127C2×#50911yescharacteristic, complemented, Sylow
153C2×#135—1271no—
154C22×D8—1271no—
155#509—1271no—
156#509—1271no—
157#509—1272no—
158#509—1272no—
159#509—1271no—
160#509D12711yescharacteristic, direct product factor
161C2×SD32—1272no—
162C2×D16—1272no—
163C2×M32—1271no—
164C2×#41—2541no—
165C22×D4—2541no—
166#135—1271no—
167#135—1271no—
168#135—1271no—
169#135—2541no—
170#135—2541no—
171#135C2×D12711yescharacteristic, complemented
172C2×Q16—1271no—
173C2×SD16—2541no—
174C2×D8—1272no—
175C2×D8—1272no—
176C2×D8—1271no—
177C2×D8—1271no—
178C2×D8—2541no—
179C2×D8—2541no—
180C2×D8—2541no—
181C2×D8C2×D12711yescharacteristic, complemented
182C22×C8—1271no—
183SD32—1272no—
184SD32—1272no—
185SD32—1272no—
186SD32C2×D12712yescomplemented
187D16—1272no—
188D16—1272no—
189D16—1272no—
190D16C2×D12712yescomplemented
191M32—1271no—
192M32—1272no—
193M32C2×D12711yescharacteristic, complemented
194C2×C16—1272no—
195C24—5081no—
196#41—21no—
197#41—5081no—
198#41—5081no—
199#41—2541no—
200#41—2541no—
201#41—2541no—
202C2×Q8—2541no—
203C2×D4—21no—
204C2×D4—5081no—
205C2×D4—5081no—
206C2×D4—5081no—
207C2×D4—5081no—
208C2×D4—2542no—
209C2×D4—2542no—
210C2×D4—2541no—
211C2×D4—2541no—
212C22×C4—1271no—
213C22×C4—5081no—
214Q16—1271no—
215Q16—2541no—
216Q16C22×D12711yescharacteristic, complemented
217SD16—21no—
218SD16—2541no—
219SD16—2541no—
220SD16—2541no—
221D8—21no—
222D8—1272no—
223D8—1271no—
224D8—2542no—
225D8—2542no—
226D8—2541no—
227D8—2541no—
228D8C22×D12712yes—
229D8C22×D12711yescharacteristic, complemented
230C2×C8—1271no—
231C2×C8—1271no—
232C2×C8—1271no—
233C2×C8—1271no—
234C2×C8—1271no—
235C2×C8C22×D12711yescharacteristic
236C16—1272no—
237C16C22×D12712yescomplemented
238C23—41no—
239C23—1271no—
240C23—10164no—
241C23—5081no—
242C23—5082no—
243C23—5082no—
244C23—5081no—
245Q8—21no—
246Q8—5081no—
247Q8—2541no—
248D4—22no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—5081no—
253D4—5082no—
254D4—5082no—
255D4—5081no—
256D4—5081no—
257D4—5081no—
258D4—5081no—
259D4—2542no—
260D4—2541no—
261C2×C4—41no—
262C2×C4—1271no—
263C2×C4—1271no—
264C2×C4—1271no—
265C2×C4—1271no—
266C2×C4—1271no—
267C2×C4—5081no—
268C2×C4—5081no—
269C2×C4—5081no—
270C2×C4—5081no—
271C2×C4D4×D12711yescharacteristic
272C8—1271no—
273C8—1271no—
274C8D4×D12711yescharacteristic
275C8C23×D12711yescharacteristic
276C22—41no—
277C22—42no—
278C22—82no—
279C22—1271no—
280C22—1271no—
281C22—10162no—
282C22—10164no—
283C22—10162no—
284C22—10164no—
285C22—5081no—
286C22—5082no—
287C22—2542no—
288C22D8×D12711yescharacteristic
289C4—41no—
290C4—1271no—
291C4—1271no—
292C4—5081no—
293C4D8×D12711yescharacteristic
294C4C2×D4×D12711yescharacteristic
295C2—21no—
296C2—82no—
297C2—81no—
298C2—1272no—
299C2—10162no—
300C2—10161no—
301C2—2541no—
302C2C2×D8×D12711yescharacteristic
subgroupG/H
centerC2C2×D8×D127
commutator, cosocleC8×C127C24
FrattiniC8C23×D127
Fitting#509×C127C2
socleC2×C127C22×D8
hypercenter#509D127
hypocenterC127C2×#509
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #135×D127C2semidirect
2. C2×D8×D127C2semidirect
3. #22550314C2semidirect
4. #22550313C2semidirect
5. #22550312C2semidirect
6. #22550311C2semidirect
7. #22550310C2semidirect
8. SD32×D127C2semidirect
9. D16×D127C2semidirect
10. M32×D127C2semidirect
11. #509×C127C2semidirect
14. #16861232C22semidirect
15. #16861231C22semidirect
18. #16861228C22semidirect
19. Q16×D127C22semidirect
21. C2×D1016C22semidirect
26. D8×D127C22semidirect
27. D8×D127C22semidirect
29. #16861154C22semidirect
30. #16861153C22semidirect
31. #16861152C22semidirect
32. D2032C22semidirect
33. #16861150C22semidirect
34. #16861150C22semidirect
35. #16861149C22semidirect
36. #16861148C22semidirect
37. C16×D127C22semidirect
38. #135×C127C22semidirect
39. C2×D8×C127C22semidirect
40. SD32×C127C22semidirect
41. D16×C127C22semidirect
42. M32×C127C22semidirect
60. Dic508C23semidirect
67. D1016C23semidirect
76. C127⋊C16C23semidirect
79. Q16×C127C23semidirect
83. D8×C127C23semidirect
85. C16×C127C23semidirect
150. D127#509direct (N × H)
152. C127C2×#509semidirect
160. #509D127direct (N × H)
171. #135C2×D127semidirect
181. C2×D8C2×D127semidirect
186. SD32C2×D127semidirect
190. D16C2×D127semidirect
193. M32C2×D127semidirect
216. Q16C22×D127semidirect
229. D8C22×D127semidirect
237. C16C22×D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#135×D1272
C2×D8×D1272
#225503142
#225503132
#225503122
#225503112
#225503102
SD32×D1272
D16×D1272
M32×D1272
#509×C1272
C2×#509127